Abstract
In this paper, we consider the dynamics of a helicoidal object, which can be either a passive particle or an active swimmer, with an arbitrary shape in a linear background flow at low Reynolds number, and derive a generalized version of the Jeffery equations for the angular dynamics of the object, including a new constant from the chirality of the object as well as the Bretherton constant. The new constant appears from the inhomogeneous chirality distribution along the axis of the helicoidal symmetry, whereas the overall chirality of the object contributes to the drift velocity. Further investigations are made for an object in a simple shear flow, and it is found that the chirality effects generate non-closed trajectories of the director vector which will be stably directed parallel or anti-parallel to the background vorticity vector depending on the sign of the chirality. A bacterial swimmer is considered as an example of a helicoidal object, and we calculate the values of the constants in the generalized Jeffery equations for a typical morphology of Escherichia coli. Our results provide useful expressions for the studies of microparticles and biological fluids, and emphasize the significance of the symmetry of an object on its motion at low Reynolds number.
Highlights
When we observe fluid flow, we sometimes perceive its motion via the motions of objects immersed in the fluid, e.g. bubbles in a stream, clouds in the sky and tracer particles in flow visualizations
The generalized Jeffery equations (3.13)–(3.14) do not seem to include the active propulsive effects of a swimmer, but note that the equations hold for an active microswimmer as well as a passive particle since the propulsive contributions, which are parallel to the director vector, are decoupled from the angular dynamics
We have considered the dynamics of a general helicoidal object, a particle or swimmer with a π/2-rotational symmetry around an axis referred to as helicoidal symmetry, in a linear background flow at low Reynolds number, and derived an equation for the director vector of the object, a generalized version of the Jeffery equation (1.1)
Summary
When we observe fluid flow, we sometimes perceive its motion via the motions of objects immersed in the fluid, e.g. bubbles in a stream, clouds in the sky and tracer particles in flow visualizations. The motions of a chiral object such as a helix in a background flow have been intensively studied in the last decade, and the coupling between the chirality and the external fluid flow has been found to generate an additional drift force, which enables us to sort the objects by the use of flow in a microdevice (Doi & Makino 2005; Makino & Doi 2005; Marcos et al 2009; Eichhorn 2010; Chen & Zhang 2011; Aristov, Eichhorn & Bechinger 2013; Hermans et al 2015; Ro, Yi & Kim 2016) This hydrodynamic coupling for a chiral object leads to biased locomotion of the bacteria in a shear flow (Marcos et al 2012) and preferential rotation of a particle in three-dimensional homogeneous isotropic turbulence (Kramel et al 2015).
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